Cut on her finger. Help!

So my sister was holding my DD yesterday and my DD leaned over and grabbed her soda can. Well my DD got her thumb in the opening and when my sister pulled her thumb out, it sliced my DDs finger. She didn't even notice or cry but it looks kinda deep. I cleaned it up last night, and put a band aid on it. She wont keep the band aid on, she just keeps biting it off. I honestly am not sure of what to put on it. I keep cleaning it because it gets dirty with fuzz (her favorite blanket). My mom says peroxide, is that okay? I don't want it to get infected.

Peroxide will keep it clean. If she won't keep a bandaid on it will be fine, air is good for it. But if you want a bandaid then over it with a sock or something, but it's not guaranteed that it will stay on either.